N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
Stationary distributions
of non Gaussian Ornstein–Uhlenbeck processes
for beam halos
CYCLOTRONS 2007 – Giardini Naxos, 1–5 October
Nicola Cufaro Petroni
Dep. of Mathematics and TIRES, Università di Bari; INFN, Sezione di Bari
S. De Martino, S. De Siena and F. Illuminati
Dep. of Physics, Università di Salerno; INFN, Sezione di Salerno
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N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
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Introduction
Popularity of Lévy processes
(Paul and Baschnagel 1999, Mantegna and Stanley 2001, Barndorff–
Nielsen et al 2001 and Cont and Tankov 2004):
from statistical physics: stable processes
(Bouchaud and Georges 1990, Metzler and Klafter 2000, Paul
and Baschnagel 1999, Woyczyński 2001),
to mathematical finance: also non stable, id processes
(Cont and Tankov 2004 and references quoted therein).
stable processes: selfsimilarity, but if non gaussian
• infinite variance (truncated distributions);
• the x decay rates of the pdf ’s can not exceed x−3 .
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infinitely divisible (id) processes: stationarity vs. selfsimilarity,
the pdf of increments could be known only at one time scale,
but new applications in the physical domain begin to emerge
(Cufaro Petroni et al 2005, 2006, Vivoli et al 2006):
the motion in the charged particle accelerator beams points
to a id, Student, Ornstein–Uhlenbeck (OU ) process.
Dynamical description: stochastic mechanics (sm)
(Nelson 1967, 1985, Guerra 1981, Guerra and Morato 1983)
suitable for many controlled, time–reversal invariant systems
(Albeverio, Blanchard and Høgh-Krohn 1983, Paul and Baschnagel
1999, Cufaro Petroni et al 1999, 2000, 2003, 2004)
Generalize sm to the non Gaussian Lévy noises:
a sm with jumps to produce halos in accelerator beams.
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Lévy processes already have applications in quantum domain:
• spinning particles (De Angelis and Jona–Lasinio 1982)
• relativistic quantum mechanics (De Angelis 1990)
• stochastic quantization (Albeverio, Rüdiger and Wu 2001).
Here: not only quantum systems, but also general complex systems (particle beams) with dynamical control.
Only one dimensional models, without going into the problem of
the dependence structure of a multivariate process
At present no Lévy sm is available: we just have OU processes
Possible underlying Lévy noises: non stable, selfdecomposable, Student processes
Student laws T (ν, δ) are not closed under convolution:
the noise distribution will not be Student at every time t.
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Results for a ν = 3 Student noise (Cufaro Petroni 2007a, 2007b):
1. for integer times t = n the noise transition law is a
mixture of a finite number of Student laws; only at
t = 1 this law is exactly T (3, δ);
2. for every finite time t the noise pdf asymptotic behavior
always is the same (x−4 ) as that of the T (3, δ) law; this
is the behavior put in evidence by Vivoli et al 2006 in the
solutions of the complex dynamical system used to study the
beams of charged particles in accelerators.
3. the stationary distribution of the OU process with T (3, δ)
noise can be calculated, and its asymptotic behavior again
is x−4 .
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Lévy processes generated by id laws
Lévy process X(t): a stationary, stochastically continuous, independent increment Markov process.
Given a type of centered, id distributions with chf ’s ϕ(au)
(a > 0), the chf of the transition law of in the time interval
[s, t] (T > 0) is
Φ(au, t − s) = [ϕ(au)](t−s)/T
(1)
and the transition pdf with initial condition X(s) = y, P-q.o.
Z M
1
p(x, t| y, s) =
lim
[ϕ(au)](t−s)/T e−i(x−y)u du
(2)
2π M →+∞ −M
N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
Along the evolution stable laws remain within the same type
and the process is selfsimilar
However, all the non gaussian stable laws:
• do not have a finite variance;
• show a rather restricted range of possible decays for large x.
Non stable, id laws have none of these shortcomings
but the Lévy processes show no selfsimilarity.
When closed under convolution:
the evolution is in the time dependence of some parameter,
but the laws do not belong to the same type.
When not even closed under convolution:
the transition laws do not remain within the same family
the evolution is not just in the time dependence of parameters.
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Student/Variance Gamma laws are id, non stable
Student laws are not even closed under convolution, but
• they are selfdecomposable
• they can have a wide range of decay laws for |x| → +∞;
• they can have finite variance
Selfdecomposable laws have two relevant properties:
1. can produce non stationary, selfsimilar, additive processes
2. alternatively can produce Lévy processes
3. are the limit laws of Ornstein–Uhlenbeck processes
When σ 2 < +∞, id Lévy processes have variance σ 2 t/T :
ordinary (non anomalous) diffusions
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Particular classes of id distributions
Variance Gamma (VG) laws VG(λ, α) (λ > 0 and α > 0):
2α
λ− 12
√ (α|x|)
fV G (x) =
Kλ− 1 (α|x|)
λ
2
2 Γ(λ) 2π
¶λ
µ
2
α
ϕV G (u) =
α 2 + u2
α is a spatial scale parameter; λ classifies different types.
VG(1, α) are the Laplace (double exponential) laws L(α)
α −α|x|
f (x) = e
,
2
α2
ϕ(u) = 2
α + u2
Asymptotic behavior of VG(λ, α) is (α|x|)λ−1 e−α|x|
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The VG laws are id, selfdecomposable, but not stable.
The VG(λ, α) with a fixed α are closed under convolution:
VG(λ1 , α) ? VG(λ2 , α) = VG(λ1 + λ2 , α)
Student laws T (ν, δ) (ν > 0, δ > 0 and B(z, w) Beta function)
fST (x) =
δB
1
¡1
µ
,
¢
ν
2 2
2
δ
δ 2 + x2
¶ ν+1
2
ν
2
(δ|u|) K ν2 (δ|u|)
¡ν ¢
ϕST (u) = 2
ν
22 Γ 2
δ is a spatial scale parameter; ν classifies different types.
Asymptotic behavior of T (ν, δ) is (|x|/δ)−ν−1
(3)
(4)
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T (1, δ) class of the Cauchy C(δ) laws
1
δ2
f (x) =
,
2
2
δπ δ +x
ϕ(u) = e−δ|u|
The Student distributions T (ν, δ) are id, selfdecomposable,
but not stable with one notable exception:
the Cauchy laws T (1, δ) = C(δ).
The Student laws are not even closed under convolution.
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The VG and Student processes
The transition chf for a VG(λ) process (α = 1, T = 1, s = 0
and y = 0) is:
¶λt
µ
1
t
(5)
Φ(u, t|λ) = [ϕV G (u)] =
2
1+u
Increment law in [0, t] is X(t) ∼ VG(λt) and the pdf is
2
λt− 12
√ |x|
Kλt− 1 (|x|)
p(x, t|λ) =
λt
2
2 Γ(λt) 2π
Asymptotic behavior (all the moments exist)
p(x, t|λ) ∼ |x|λt−1 e−|x| ,
|x| → +∞
(6)
N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
The Student family T (ν, δ) is not closed under convolution
An explicit form of the transition pdf not available
We study the ν = 3, T (3, δ) Student process candidate to describe the increments in the velocity process for particles in an
accelerator beam (Vivoli et al 2006).
For δ = 1, the T (3, 1)–process has transition pdf
½ t+ix
¾
e
Γ(t + 1, t + ix)
p(x, t| 3) = <
(7)
t+1
π(t + ix)
with Γ(a, z) the incomplete Gamma function, and
¡ −4 ¢
2t
p(x, t| 3) =
+ o |x|
,
|x| → +∞
(t > 0)
4
πx
For fixed, finite t > 0 the asymptotic behavior of p(x, t| 3) is
always infinitesimal at the same order |x|−4 of the original T (3, 1)
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At integral times t = n = 1, 2, . . . the transition pdf p(x, n| 3) of
the T (3, 1)–Student process is a mixture of Student pdf ’s with
• odd integer orders ν = 2k + 1 with k = 0, 1, . . .,
• integer scaling factors δ = n,
• relative weights
k 2k+1
Xµ
(−1)
qn (k| 3) =
2k + 1
j=0
¶µ
¶µ ¶
µ ¶j
n 2k + 1
j
−1
(j + 1)!
2n
j
j
k
namely: mixtures of T (2k + 1, n) laws with k = 1, . . . , n with no
Student law of order smaller than ν = 3
The distributions qn (k| 3) are displayed in Figure 1.
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n=5
0.3
pn HkÈ 3L
0.25
n = 10
0.2
0.15
n = 20
0.1
n = 40
0.05
k
10
20
30
40
Figure 1: Mixture weights of the integer time (t = n) components
for a Student process with ν = 3.
N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
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Ornstein–Uhlenbeck processes
The VG and Student processes have no Brownian component:
they are pure jump processes
The VG and Student processes have infinite activity: namely the
set of jump times is countably infinite and dense in [0, +∞].
At first sight the simulated samples of both a VG and a Student process do not look very different from that of a Wiener
process.
Lévy diffusions Y (t):
solutions of SDE driven by a Lévy process X(t)
dY (t) = α(t, Y (t)) dt + dX(t)
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Compare Ornstein–Uhlenbeck (OU) processes
driven by either VG or T (3, δ) noises X(t)
dY (t) = −b Y (t) dt + dX(t)
(8)
with usual OU process driven by Brownian noise B(t)
dY (t) = −b Y (t) dt + dB(t)
Figure 2: samples of 5 000 steps with noise laws in the Table
(a)
N (0, 1)
√1
2π
−x2 /2
e
(b)
√
VG(1, 2)
√1
2
e
√
− 2 |x|
(c)
T (3, 1)
2
1
π (1+x2 )2
(9)
N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
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10
5
HaL
15
10
5
HbL
Τ
-5
-10
-15
-20
15
10
5
-5
-10
-15
-20
HcL
Τ
-5
-10
-15
-20
Τ
100
80
60
40
20
HdL
Τ
Figure 2: OU processes driven by (a) Brownian motion; (b) VG Lévy
noise; (c) Student Lévy noise; and then (d) Student OU–type process
with restoring force of finite range.
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(a) typical OU process driven by normal Brownian motion,
(b) and (c) OU–type processes driven by VG and Student noises
(a) is rather strictly confined by the restoring force −by
(b) and (c) show spikes going outside the confining region.
(d) take a restoring force of a finite range:
dY (t) = α(Y (t)) dt + dX(t)

 −by, for |y| ≤ q;
α(y) =
 0,
for |y| > q.
q>0
The restoring force acts only in [−q, q]
When the process jumps beyond y = ±q it diffuses freely:
possible model of halo formation in particle beams
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Role of selfdecomposability in OU processes:
if Y (t) is solution of the OU SDE
dY (t) = −b Y (t) dt + dX(t)
for a Lévy noise X(t) with logarithmic characteristic ψ(u) =
log ϕ(u), then the stationary distribution is absolutely continuous
and selfdecomposable with logarithmic characteristic ψY (u) such
that
Z
∞
ψY (u) =
0
ψ(ue−bt ) dt ,
ψ(u) = buψY0 (u)
VG and Student laws are id and selfdecomposable
Then we can explicitly find the stationary laws of the OU processes with VG and Student noises.
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√
OU stationary distribution for a Laplace law VG(1, 2/a)
2
with variance σX
= a2 :
µ
¶
µ 2 2¶
2 2
au
au
1
ψ(u) = − log 1 +
Li2 −
,
ψY (u) =
2
2b
2
where dilogarithm is
Z 0
log(1 − s)
Li2 (x) =
ds
s
x
Ã
=
∞
X
k=1
variance of the stationary distribution
2
a
σY2 = −ϕ00Y (0) =
2b
pdf can be numerically evaluated
xk
,
2
k
!
|x| ≤ 1
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0.8
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
-10
-5
5
10
-3
-2
-1
1
2
3
Figure 3: chf and pdf of a OU stationary distribution (black lines),
√
compared with the chf and pdf of the driving VG(1, 2/a) Laplace
noise (red lines). Here a = b = 1.
N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
OU stationary distribution for a student law T (3, a)
2
with variance σX
= a2 :
ψ(u) = −a|u| + log(1 + a|u|) ,
a|u| 1
ψY (u) = −
− Li2 (−a|u|)
b
b
variance of the stationary distribution
2
a
σY2 = −ϕ00Y (0) =
2b
pdf f (x) can be numerically evaluated and
f (x) ∼ 0.4244 × x−4 ,
x→∞
namely: the stationary solution is not a Student law, but it keeps
the same asymptotic behavior of the driving Student noise.
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0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
-10
-5
5
10
-3
-2
-1
1
2
3
Figure 4: chf and pdf of a OU stationary distribution (black lines),
compared with the chf and pdf of the driving T (3, a) Student noise
(red lines). Here a = b = 1.
N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
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Conclusions
• An OU process driven by a selfdecomposable T (3, a) Student noise seems to be a good candidate as a model for halo
formation in beams of charged particles in accelerators
• The driving Student noise and the stationary laws show the
same asymptotic behavior (x−4 ) of dynamical simulations
• Selfdecomposable processes are at present under intense scrutiny
for possible use in option pricing (Carr et al 2007)
• A dynamical model (SM) for processes driven by non Gaussian Lévy noises must now be elaborated in order to achieve
a reasonable control of the beam size.
N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
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References
Albeverio S, Blanchard P and Høgh-Krohn R 1983 Expo. Math.
4 365
Albeverio S, Rüdiger B and Wu J–L 2001 in Lévy processes,
Theory and applications ed Barndorff–Nielsen O et al (Boston:
Birkhäuser) p 187
Barndorff–Nielsen O E 2000 Probability densities and Lévy densities (MaPhySto, Aarhus, Preprint MPSRR/2000-18)
Bouchaud J–P and Georges A 1990 Phys. Rep. 195 127
Carr P, Geman H, Madan D and Yor M 2007 Math. Fin. 17 31
Cont R and Tankov P 2004 Financial modelling with jump processes (Boca Raton: Chapman&Hall/CRC)
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N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
Cufaro Petroni N, De Martino S, De Siena S and Illuminati F
1999 J. Phys. A 32 7489
Cufaro Petroni N, De Martino S, De Siena S and Illuminati F
2000 Phys. Rev. E 63 016501
Cufaro Petroni N, De Martino S, De Siena S and Illuminati F
2003 Phys. Rev. ST Accelerators and Beams 6 034206
Cufaro Petroni N, De Martino S, De Siena S and Illuminati F
2004 Int. J. Mod. Phys. B 18 607
Cufaro Petroni N, De Martino S, De Siena S and Illuminati F
2005 Phys. Rev. E 72 066502
Cufaro Petroni N, De Martino S, De Siena S and Illuminati F
2006 Nucl. Instr. Meth. A 561 237
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Cufaro Petroni N 2007a J. Phys. A 40 2227
Cufaro Petroni N 2007b Selfdecomposability and selfsimilarity: a
concise primer arXiv:0708.1239v2 [cond-mat.stat-mech]
De Angelis G F 1990 J. Math. Phys. 31 1408
De Angelis G F and Jona–Lasinio G 1982 J. Phys. A 15 2053
Guerra F 1981 Phys. Rep. 77 263
Guerra F and Morato L M 1983 Phys. Rev. D 27 1774
Mantegna R and Stanley H E 2001 An introduction to econophysics (Cambridge: Cambridge University Press)
Metzler R and Klafter J 2000 Phys. Rep. 339 1
Nelson E 1967 Dynamical theories of Brownian motion (Princeton: Princeton University Press)
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N Cufaro Petroni: CYCLOTRONS 2007 – Giardini Naxos, 1–5 October, 2007
Nelson E 1985 Quantum Fluctuations (Princeton: Princeton University Press)
Paul W and Baschnagel J 1999 Stochastic processes: from physics
to finance (Berlin: Springer)
Vivoli A, Benedetti C and Turchetti G 2006 Nucl. Instr. Meth.
A 561 320
Woyczyński W A 2001 in Lévy processes, Theory and applications
ed Barndorff–Nielsen O et al (Boston: Birkhäuser) p 241
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